As noted before, two sixth order compact stencils have to be developed for 2nd and 4th order derivative terms to solve the 1D CDE using the Lax-Wendroff method in Eq. (REF ). The rationale for such a choice is due to the proven accuracy of NUC6 scheme [1]}, and also because the stencils for 1st, 2nd and 4th order derivatives on uniform grids reduce to the well known sixth order interior stencils of schemes proposed by Lele [2]}.
For a general non-uniform structured grid in 1D with \(M\)  points, the stencils for computing the first, second and fourth derivatives are given as:
\(\begin{split}&\text{First Derivative:}\\&u^{\prime }_1=\frac{1}{h_{r2}}\left[\left(-(\beta _2+1)+\frac{1}{\beta _2+1} \right)u_1 + (\beta _2+1)u_2 -\frac{u_3}{\beta _2+1}\right]\\[1.5ex]&\frac{u^{\prime }_1}{2} + u^{\prime }_2 + \frac{u^{\prime }_3}{2} = \frac{u_2-u_1}{h_{l2}} + \frac{u_3-u_2}{h_{r2}}\\[1.5ex]&p^1_{j-1}u^{\prime }_{j-1} + p^1_{j}u^{\prime }_{j} + p^1_{j+1}u^{\prime }_{j+1}=s^1_{-2}(u_j-u_{j-2}) + s^1_{-1}(u_j-u_{j-1})+ s^1_{1}(u_{j+1}-u_j)\\&\hspace{0.0pt}+ s^1_{2}(u_{j+2}-u_j)\\[1.5ex]&\frac{u^{\prime }_{M-2}}{2} + u^{\prime }_{M-1} + \frac{u^{\prime }_M}{2} = \frac{u_{M-1}-u_{M-2}}{h_{lM-1}} + \frac{u_M-u_{M-1}}{h_{rM-1}}\\[1.5ex]&u^{\prime }_M=\frac{1}{h_{rM-1}}\left[\left((\beta _{M-1}+1)-\frac{\beta ^2_{M-1}}{\beta _{M-1}+1} \right)u_M - (\beta _{M-1}+1)u_{M-1} +\frac{\beta ^2_{M-1} u_{M-2}}{\beta _{M-1}+1}\right]\\\end{split}\) 
\(\begin{split}&\text{Second Derivative:}\\\hspace{2.84526pt}&u^{\prime \prime }_1=\frac{2(h_{r1}+h_{rr1}+h_{rrr1})}{h_{r1}h_{rr1}h_{rrr1}}u_1 - \frac{2(h_{rr1}+h_{rrr1})}{h_{r1}(h_{r1}-h_{rrr1})(h_{r1}-h_{rr1})}u_2\\\hspace{2.84526pt}&\hspace{28.45274pt} - \frac{2(h_{r1}+h_{rrr1})}{h_{rr1}(h_{r1}-h_{rr1})(h_{rrr1}-h_{rr1})}u_3 + \frac{2(h_{r1}+h_{rr1})}{h_{rrr1}(h_{r1}-h_{rrr1})(h_{rrr1}-h_{rr1})}u_4\\[1.5ex]\hspace{2.84526pt}&u^{\prime \prime }_2 = \frac{2}{h_{l2}(h_{l2}+h_{r2})}u_1 - \frac{2}{h_{l2}h_{r2}}u_2 + \frac{2}{h_{r2}(h_{l2}+h_{r2})}u_3\\[1.5ex]\hspace{2.84526pt}&p^2_{j-1}u^{\prime \prime }_{j-1} + p^2_{j}u^{\prime \prime }_{j} + p^2_{j+1}u^{\prime \prime }_{j+1}=s^2_{-2}u_{j-2} + s^2_{-1}u_{j-1}+ s^2_0 u_j + s^2_{1}u_{j+1}+ s^2_{2}u_{j+2}\\[1.5ex]\hspace{2.84526pt}&u^{\prime \prime }_{M-1} = \frac{2}{h_{lM-1}(h_{lM-1}+h_{rM-1})}u_{M-2} - \frac{2}{h_{lM-1}h_{rM-1}}u_{M-1} + \frac{2}{h_{rM-1}(h_{lM-1}+h_{rM-1})}u_M\\[1.5ex]\hspace{2.84526pt}&u^{\prime \prime }_M=\frac{2(h_{lM}+h_{llM})}{h_{lllM}(h_{lM}-h_{lllM})(h_{lllM}-h_{llM})}u_{M-3} - \frac{2(h_{lM}+h_{lllM})}{h_{llM}(h_{lM}-h_{llM})(h_{lllM}-h_{llM})}u_{M-2} \\\hspace{2.84526pt}&\hspace{28.45274pt} - \frac{2(h_{lllM}+h_{llM})}{h_{lM}(h_{lM}-h_{lllM})(h_{lM}-h_{llM})}u_{M-1}+ \frac{2(h_{lM}+h_{llM}+h_{lllM})}{h_{lM}h_{llM}h_{lllM}}u_M\\\end{split}\) 
\(\begin{split}&\text{Fourth Derivative:}\\&u^{\prime \prime \prime \prime }_1 = \frac{24}{h_{r1}h_{rr1}h_{rrr1}h_{rrrr1}}u_1 + \frac{24}{h_{r1}(h_{r1}-h_{rrr1})(h_{r1}-h_{rrrr1})(h_{r1}-h_{rr1})}u_2 \\&\hspace{28.45274pt}-\frac{24}{h_{rr1}(h_{r1}-h_{rr1})(h_{rrr1}-h_{rr1})(h_{rrrr1}-h_{rr1})}u_3\\&\hspace{28.45274pt}-\frac{24}{h_{rrr1}(h_{r1}-h_{rrr1})(h_{rrr1}-h_{rrrr1})(h_{rrr1}-h_{rr1})}u_4 \\&\hspace{28.45274pt}+\frac{24}{h_{rrrr1}(h_{r1}-h_{rrrr1})(h_{rrr1}-h_{rrrr1})(h_{rrrr1}-h_{rr1})}u_5\\[1.5ex]&u^{\prime \prime \prime \prime }_2 = \frac{24}{h_{l2}(h_{l2} + h_{r2})(h_{l2}+h_{rrr2})(h_{l2}+h_{rr2})}u_1 - \frac{24}{h_{l2}h_{r2}h_{rr2}h_{rrr2}}u_2 \\&\hspace{28.45274pt}+\frac{24}{h_{r2}(h_{l2}+h_{r2})(h_{r2}-h_{rrr2})(h_{r2}-h_{rr2})}u_3 + \frac{24}{h_{rr2}(h_{l2}+h_{rr2})(h_{r2}-h_{rr2})(h_{rrr2}-h_{rr2})}u_4 \\&\hspace{28.45274pt}- \frac{24}{h_{rrr}(h_{l}+h_{rrr})(h_{r}-h_{rrr})(h_{rrr}-h_{rr})}u_5\\[1.5ex]&p^4_{j-1}u^{\prime \prime \prime \prime }_{j-1} + p^4_{j}u^{\prime \prime \prime \prime }_{j} + p^4_{j+1}u^{\prime \prime \prime \prime }_{j+1}=s^4_{-2}u_{j-2} + s^4_{-1}u_{j-1}+ s^4_0 u_j + s^4_{1}u_{j+1}+ s^4_{2}u_{j+2}\\[1.5ex]&u^{\prime \prime \prime \prime }_{M-1} = -\frac{24}{h_{lllM-1}(h_{lllM-1}+h_{rM-1})(h_{lM-1}-h_{lllM-1})(h_{lllM-1}-h_{llM-1})}u_{M-4} \\&\hspace{28.45274pt}+ \frac{24}{h_{llM-1}(h_{llM-1}+h_{rM-1})(h_{lM-1}-h_{llM-1})(h_{lllM-1}-h_{llM-1})}u_{M-3} \\&\hspace{28.45274pt}+\frac{24}{h_{lM-1}(h_{lM-1}+h_{rM-1})(h_{lM-1}-h_{lllM-1})(h_{lM-1}-h_{llM-1})}u_{M-2}\\&\hspace{28.45274pt}-\frac{24}{h_{lM-1}h_{llM-1}h_{lllM-1}h_{rM-1}}u_{M-1}\\&\hspace{28.45274pt}+ \frac{24}{h_{r}(h_l + h_r)(h_{lll}+h_r)(h_{ll}+h_r)}u_M\\[1.5ex]&u^{\prime \prime \prime \prime }_M = \frac{24}{h_{llllM}(h_{lM}-h_{llllM})(h_{lllM}-h_{llllM})(h_{llllM}-h_{llM})}u_{M-4}\\&\hspace{28.45274pt}-\frac{24}{h_{lllM}(h_{lM}-h_{lllM})(h_{lllM}-h_{llllM})(h_{lllM}-h_{llM})}u_{M-3}\\&\hspace{28.45274pt}-\frac{24}{h_{llM}(h_{lM}-h_{llM})(h_{lllM}-h_{llM})(h_{llllM}-h_{llM})}u_{M-2}\\&\hspace{28.45274pt}+\frac{24}{h_{lM}(h_{lM}-h_{lllM})(h_{lM}-h_{llllM})(h_{lM}-h_{llM})}u_{M-1} \\&\hspace{28.45274pt}+\frac{24}{h_{lM}h_{llM}h_{lllM}h_{llllM}}u_M \\\end{split}\) 
